Central Limit Theorem Calculator — Sampling Distribution of x̄
Enter the population mean, standard deviation, and sample size to see the sampling distribution of the sample mean (x̄). The CLT guarantees this distribution is approximately normal for large n, regardless of the original population shape.
SE = σ / √n — spread of the sampling distribution of x̄
- 1
Square root of sample size (√n)
√30 = 5.4772 - 2
Standard error SE = σ ÷ √n
10 ÷ 5.4772 = 1.8257SE shrinks as n grows — doubling the sample size halves the standard error.
How does this calculator work?
The CLT says sample means follow N(μ, σ²/n) for large enough n. Standard error SE = σ/√n shrinks with increasing sample size — doubling n halves SE. Use n ≥ 30 as a rule of thumb. Interval widths: 68% within ±1 SE, 95% within ±1.96 SE, 99.7% within ±3 SE.
Formula
How this is calculated
The Central Limit Theorem (CLT) states that, given a population with any distribution having mean μ and finite standard deviation σ, the distribution of sample means from repeated samples of size n approaches a normal distribution as n grows. For n ≥ 30, the approximation is generally reliable; for symmetric or near-normal populations it holds even for n as small as 10–15.
The spread of this sampling distribution is measured by the standard error of the mean: SE = σ / √n. As sample size grows, SE shrinks — doubling n halves the SE. The sampling distribution is centred exactly at μ, so the expected value of the sample mean is always the population mean. Around this centre, 68% of sample means fall within ±1 SE, 95% within ±1.96 SE, and 99.7% within ±3 SE — the empirical rule applied to the sampling distribution.
This calculator assumes σ is the known population standard deviation. In practice σ is usually estimated from sample data as s, and the t-distribution provides better coverage for small n. The CLT result underlies nearly all classical hypothesis tests and confidence intervals for means.
Frequently asked questions
The CLT says that the distribution of sample means from a large enough sample will be approximately normal, no matter what shape the original population has. This allows us to apply normal-distribution probability tools to draw inferences about population means.
A commonly used rule of thumb is n ≥ 30. For populations already roughly symmetric or normal, n as small as 10–15 may suffice. For heavily skewed populations you may need n > 50 or more before the sampling distribution is well-approximated by normal.
The standard error (SE = σ/√n) is the standard deviation of the sampling distribution of x̄. It measures how much sample means vary from sample to sample. A smaller SE means your sample mean is a more precise estimate of μ — achieved by increasing n or reducing population variability.
Also known as
TG we-Calculate Editorial Team. (2026). Central Limit Theorem Calculator — Sampling Distribution of x̄ [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/central-limit-theorem-calculator
TG we-Calculate Editorial Team. "Central Limit Theorem Calculator — Sampling Distribution of x̄." TG we-Calculate. 2026. https://we-calculate.com/calculator/central-limit-theorem-calculator.
TG we-Calculate Editorial Team, "Central Limit Theorem Calculator — Sampling Distribution of x̄," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/central-limit-theorem-calculator
@misc{wecalculate_central_limit_theorem_calculator, title = {Central Limit Theorem Calculator — Sampling Distribution of x̄}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/central-limit-theorem-calculator}}, year = {2026}, note = {TG we-Calculate} }
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